I have read the definitions and basics of random graphs, but I don't really understand what is a random graph, and what's a "non-random" graph. A non mathematical explanation would be of great help.
Let $A01, A02, ... , A20$ be labels of $20$ people in an experiment. In the graph theory labels are equal to nodes. Suppose the $20$ people were allowed to nominate each other, these nominations let be edges.
Figure illustrates the original graph $G$ reflecting the structure of nominations. The graph $G$ is non-random because it was built on nominations.
We can compute different graph statistics, for instance, diameter. It’s seen that the diameter of graph $G$ equal to $6$ because the "longest shortest path" (green on Figure) between members $A15$, $A06$, $A12$, $A19$, $A11$, $A08$ and $A10$ takes $6$ edges.
The simulation of the random graphs can be used to evaluate the significance of the statistics. Based on topological properties (number of nodes, numbers of edges) of the original graph $G$ $1000$ random graphs were generated to compute their average diameter. These $1000$ graphs are random graphs. Btw, in my case the average diameter equals to $6$.